Trigonometry Examples

$\frac{4 \sqrt{3}}{\sqrt{10}}$

Step 1

Multiply $\frac{4 \sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{4 \sqrt{3}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

Multiply $\frac{4 \sqrt{3}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{4 \sqrt{3} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 2.1.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{4 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 2.1.3

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{4 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 2.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 2.1.5

Add $1$ and $1$ .

$\frac{4 \sqrt{3} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 2.1.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{4 \sqrt{3} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 2.1.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 \sqrt{3} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 2.1.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{4 \sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{3} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.1.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{3} \sqrt{10}}{10^{1}}$

Step 2.1.6.5

Evaluate the exponent.

$\frac{4 \sqrt{3} \sqrt{10}}{10}$

Step 2.2

Cancel the common factor of $4$ and $10$ .

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Step 2.2.1

Factor $2$ out of $4 \sqrt{3} \sqrt{10}$ .

$\frac{2 (2 \sqrt{3} \sqrt{10})}{10}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $2$ out of $10$ .

$\frac{2 (2 \sqrt{3} \sqrt{10})}{2 (5)}$

Step 2.2.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{3} \sqrt{10})}{2 \cdot 5}$

Step 2.2.2.3

Rewrite the expression.

$\frac{2 \sqrt{3} \sqrt{10}}{5}$

Step 3

Simplify the numerator.

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Step 3.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{10 \cdot 3}}{5}$

Step 3.2

Multiply $10$ by $3$ .

$\frac{2 \sqrt{30}}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{30}}{5}$

Decimal Form:

$2.19089023 \dots$